Statistical Methods

The term "degrees of freedom" comes from the idea that if we have set number of scores (N) that add up to a certain sum, then only N -1 scores are free to vary.

The last must be the number that would make all the scores add up to the sum.

The reason we use degrees of freedom as a divisor in calculating sample variance is that it yields an unbiased estimator of the population variance.

A sample statistic is unbiased if the average value of the sample statistic, obtained over many different samples, is equal to the population parameter.

If the average value of a sample statistic overestimates or underestimates the population parameter, it is said to be biased.

Why do we use variance at all if the SD is much easier to understand?

Because the variance is an unbiased estimate of the population variance, while the SD is a biased estimator of the sample SD.

The mean is an unbiased estimate of the population mean and needs no adjustment.

The mean and the variance give us the best unbiased estimates of the population parameters.

z-scores are a statistical technique that allow us to transform each score in a distribution into a standard score.

Standard scores take the scale of measurement out of the picture and make it easier to compare scores with each other.

The position of one score (say you had a 70 on a stats quiz) is "good" or "bad" relative to the mean.

The relative location of a score depends on the mean and standard deviation as well as the actual score.

Ex: You make a 70 on an exam with a mean of 65. If the standard deviation is 10, you did just above average. If the standard deviation is 2, you did great. (see figure 5.1 on p. 105).

Standardizing takes different distributions and makes them equivalent.

IQ tests are a good example: we know 145 is very high even if we don’t know which particular IQ test was taken. Mean = 100 SD = 15

z-scores are a signed number: if the z-score is negative, than the score is below the mean, if it is positive, it is above the mean

The number tells the distance from the mean in standard deviation units

IQ example: 100, z = 0; 115, z = 1; 130, z = 2; 145, z = 3.

A distribution has a mean of 100 and a standard deviation of 10. What z-score corresponds to a score of X = 120?

Z = 2.00

Multiply the z-score by the standard deviation

Z(SD)

If we change every raw score in a distribution into a z-score, we create a distribution of z-scores.

This new distribution has characteristics that make it very useful

These properties deal with the shape, mean, and standard deviation of the distribution

The shape is the same as the raw scores

Skewed distributions stat skewed, normal distributions stay normal

Each score’s position (rank order) remains the same

The z-score distribution will always have a mean of zero

The original population mean is transformed into a value of zero in the z-score distribution.

This mean of zero makes it easy to identify locations. (-,0,+)

The distribution of z-scores will always have a SD of 1.

We know the exact distance of a score from the mean

Because the values of the mean and standard deviation are always the same, they can be used to make dissimilar distributions comparable.

We can compare different scores even if they come from different distributions

Many people don’t like to use z distributions because the negative numbers and decimals can be distracting.

We can create a standardized distribution based on z that has whole, positive numbers.

The idea is to compute z, then transform z such that some specific mean and standard deviation is obtained—like with IQ scores.

[Ignore the book on this—its confusing]

All you have to do is take the z score, multiply it by the standard deviation of the new scale, and then add the mean of the new scale

A popular transformed standard score is known a the T-score. It has a mean of 50 and a SD of 10. It is calculated as follows:

T = (z)(10)+50

We can also standardize by specifying any mean and standard deviation: such as IQ tests:

IQ = (Z)(15)+100